The correct answer is $\boxed{\text{B) }35}$.
The value of the integral of a function $g(x, y)$ along a straight line segment from the point $(x_0, y_0)$ to the point $(x_1, y_1)$ is given by the formula
$$\int_{x_0}^{x_1} g(x, y) dx = \left[ g(x, y) \right]_{x_0}^{x_1} = g(x_1, y_1) – g(x_0, y_0).$$
In this case, we have $g(x, y) = 4x^3 + 10y^4$, $x_0 = 0$, $x_1 = 1$, $y_0 = 0$, and $y_1 = 2$. Substituting these values into the formula, we get
$$\int_{0}^{1} 4x^3 + 10y^4 dx = \left[ 4x^4 + 5x^2y^4 \right]_{0}^{1} = 4 + 10 \cdot 2^4 – 0 = \boxed{35}.$$
Option A is incorrect because it is the value of the integral of $g(x, y)$ along the straight line segment from the point $(0, 0)$ to the point $(1, 1)$. Option C is incorrect because it is the value of the integral of $g(x, y)$ along the straight line segment from the point $(0, 0)$ to the point $(2, 0)$. Option D is incorrect because it is the value of the integral of $g(x, y)$ along the straight line segment from the point $(0, 0)$ to the point $(1, 0)$.